
Mathematical Biosciences and Engineering, 2019, 16(1): 5677. doi: 10.3934/mbe.2019003
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Dynamics of a delay turbidostat system with contois growth rate
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, PR China
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References
1. R. Arditi and L.R. Ginzburg, Coupling in predatorprey dynamics: ratiodependence, J. Theor. Biol., 139 (1989), 311–332.
2. A.W. Bush and A.E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol., 63 (1975), 385–396.
3. J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192.
4. D.E. Contois, Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures, J. Gen. Microbiol., 21 (1959), 40–50.
5. J. Flegr, Two distinct types of natural selection in turbidostatlike and chemostatlike ecosystems, J. Theor. Biol., 188 (1997), 121–126.
6. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983.
7. B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
8. D. Herbert, R. Elsworth and B.C. Telling, The continuous culture of bacteria; a theoretical and experimental study, J. Gen. Microbiol., 14 (1956), 601–622.
9. S.B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for singlenutrient competition in continuous culture of microorganism, SIAM J. Appl. Math., 32 (1997), 366–383.
10. Z.X. Hu, G.K. Gao and W.B. Ma, Dynamics of a threespecies ratiodependent diffusive model, Nonlinear Anal. Real, 11 (2010), 2106–2114.
11. X.Y. Hu, Z.X. Li and X.G. Xiang, Feedback control for a turbidostat model with ratiodependent growth rate, J. Appl. Math. Inform., 31 (2013), 385–398.
12. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
13. P.D. Leenheer and H. Smith, Feedback control for the chemostat, J. Math. Biol., 46 (2003), 48–70.
14. B.T. Li, Competition in a turbidostat for an inhibitory nutrient, J. Biol. Dynam., 2 (2008), 208–220.
15. B.T. Li, G.S.K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), 2058–2086.
16. Z.X. Li and L.S. Chen, Periodic solution of a turbidostat model with impulsive state feedback control, Nonlinear Dynam., 58 (2009), 525–538.
17. Z. Li and R. Xu, Stability analysis of a ratiodependent chemostat model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243–253.
18. N. MacDonald, Time lag in simple chemostat models, Biot. echnol. Bioeng., 18 (1976), 805–812.
19. J. Monod, The growth of bacterial culture, Annu. Rev. Microbiol., 3 (1949), 371–394.
20. H. Moser, The dynamics of bacterial populations maintained in the chemostat, Cold Spring Harb. Sym., 22 (1957), 121.
21. M.I. Nelson and H.S. Sidhu, Reducing the emission of pollutants in food processing wastewaters, Chem. Eng. Process., 46 (2007), 429–436.
22. R.T. Alqahtani, M.I. Nelson and A.L. Worthy, Analysis of a chemostat model with variable yield coefficient and substrate inhibition: contois growth kinetics, Chem. Eng. Process., 202 (2015), 332–344.
23. Z.C. Jiang andW.B. Ma, Delayed feedback control and bifurcation analysis in a chaotic chemostat system, Int. J. Bifurcat. Chaos, 25 (2015), 1550087.
24. C. Jost, Predatorprey theory: hidden twins in ecology and microbiology, Oikos, 90 (2000), 202–208.
25. S.G. Ruan and J.J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Cont. Discrete Impul. syst. Ser. A., 10 (2003), 863–874.
26. S.G. Ruan and G.S.K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), 786–812.
27. H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springe, New York, 2010.
28. O. Tagashira and T. Hara, Delayed feedback control for a chemostat model, Math. Biosci., 201 (2006), 101–112.
29. Y. Tian, Y. Bai and P. Yu, Impact of delay on HIV1 dynamics of fighting a virus with another virus, Math. Biosci. Eng., 11 (2014), 1181–1198.
30. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, UK, 1995.
31. L.Wang and G.S.K.Wolkowicz, A delayed chemostat model with general nonmonotone response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452–468.
32. Y. Yao, Z.X. Li and Z.J. Liu, Hopf bifurcation analysis of a turbidostat model with discrete delay, Appl. Math. Comput., 262 (2015), 267–281.
33. S.L. Yuan, P. Li and Y.L. Song, Delay induced oscillations in a turbidostat with feedback control, J. Math. Chem., 49 (2011), 1646–1666.
34. Z. Zhao and X.Y. Song, Bifurcation and complexity in a ratiodependent predatorprey chemostat with pulsed input, Appl. Mat. Ser. B, 22 (2007), 379–387.
35. Z.F. Zhang, T.R. Ding, W.Z. Huang and Z.X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI., 1992.
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